Enumeration of lozenge tilings of a hexagon with a shamrock missing on the symmetry axis

TL;DR

This paper proves a simple product formula for counting lozenge tilings of symmetric hexagons with a shamrock-shaped hole placed at any position along the symmetry axis, extending previous specific cases.

Contribution

It generalizes existing results by providing a universal tiling enumeration formula for symmetric hexagons with a shamrock hole at any symmetry axis position.

Findings

A closed-form product formula for tilings with shamrock at any symmetry axis position.

Extension of Eisenk"olbl's work on hexagons with two missing triangles.

Unified enumeration result for symmetric hexagons with shamrock holes.

Abstract

In their paper about a dual of MacMahon's classical theorem on plane partitions, Ciucu and Krattenthaler proved a closed form product formula for the tiling number of a hexagon with a "shamrock", a union of four adjacent triangles, removed in the center (Proc. Natl. Acad. Sci. USA 2013). Lai later presented a $q$-enumeration for lozenge tilings of a hexagon with a shamrock removed from the boundary (European J. Combin. 2017). It appears that the above are the only two positions of the shamrock hole that yield nice tiling enumerations. In this paper, we show that in the case of symmetric hexagons, we always have a simple product formula for the number of tilings when removing a shamrock at any position along the symmetry axis. Our result also generalizes Eisenk\"olbl's related work about lozenge tilings of a hexagon with two unit triangles missing on the symmetry axis (Electron. J. Combin. 1999).